ISC Practicals and Theory - Kaprekar Number

A Kaprekar number is a number whose square, when split into two parts, gives parts that add up to the number itself. This program is written in the main method alone: it counts the digits, squares the number, splits the square and compares the sum of the two parts with the number.

Definition

Let $n$ be a number with $d$ digits. Find $n^2$ and split it into a right part of exactly $d$ digits and a left part made of the remaining digits (the left part is 0 if there are no remaining digits). If left part + right part $= n$, then $n$ is a Kaprekar number.

• $45$ has 2 digits: $45^2 = 2025$, right part $= 25$, left part $= 20$, and $20 + 25 = 45$. So 45 is a Kaprekar number.
• $9$ has 1 digit: $9^2 = 81$, right part $= 1$, left part $= 8$, and $8 + 1 = 9$. So 9 is a Kaprekar number.
• $297$ has 3 digits: $297^2 = 88209$, right part $= 209$, left part $= 88$, and $88 + 209 = 297$. So 297 is a Kaprekar number.

This program follows the usual rule that the right part must not be zero. Because of this, 10 is not a Kaprekar number: $10^2 = 100$, right part $= 00 = 0$, left part $= 1$, and $1 + 0 = 1$, which is not 10. (Without this rule, 10, 100, 1000 and so on would wrongly pass, since $10^2 = 100$ splits as 10 + 00.) The number 1 is a Kaprekar number: $1^2 = 1$, right part $= 1$, left part $= 0$, and $0 + 1 = 1$.

Algorithm

1. Accept the number n. If n <= 0, display an invalid input message and stop.
2. Copy n to temp and count its digits in digits: while temp > 0, add 1 to digits and do temp = temp / 10.
3. Find the square sq = (long) n * n. The long type is used because the square of a large int does not fit in an int.
4. Start with div = 1 and multiply it by 10 exactly digits times, so that div becomes $10^{digits}$.
5. Split the square: right = sq % div is the last digits digits and left = sq / div is the remaining digits.
6. If right > 0 and left + right == n, display the square, the two parts and that n is a Kaprekar number. Otherwise display that it is not.

Program

Everything is written inside main: the digit count, the square, the power of 10, the split and the final test.

import java.util.Scanner;
public class Kaprekar
{
    public static void main(String args[])
    {
        Scanner sc = new Scanner(System.in);
        System.out.print("Enter a number: ");
        int n = sc.nextInt();                       // the number to be checked
        if (n <= 0)
        {
            System.out.println("Invalid input. Enter a positive number.");
        }
        else
        {
            int temp = n, digits = 0;               // digits counts the digits of n
            while (temp > 0)
            {
                digits++;
                temp = temp / 10;
            }
            long sq = (long) n * n;                 // the square of n
            long div = 1;                           // 10 raised to digits
            for (int i = 1; i <= digits; i++)
            {
                div = div * 10;
            }
            long right = sq % div;                  // the last digits of the square
            long left = sq / div;                   // the remaining digits of the square
            if (right > 0 && left + right == n)
            {
                System.out.println("Square = " + sq);
                System.out.println("Left part = " + left + ", Right part = " + right);
                System.out.println(n + " is a Kaprekar number.");
            }
            else
                System.out.println(n + " is not a Kaprekar number.");
        }
    }
}

Variable description table

Sample input and output

Enter a number: 45
Square = 2025
Left part = 20, Right part = 25
45 is a Kaprekar number.

Enter a number: 10
10 is not a Kaprekar number.

Enter a number: 1
Square = 1
Left part = 0, Right part = 1
1 is a Kaprekar number.

Enter a number: -5
Invalid input. Enter a positive number.

Dry run

Take n = 45. The counting loop gives digits = 2. Then the program continues as follows.

Both conditions are true, so the program prints the square, the two parts and that 45 is a Kaprekar number.

For the counting loop the trace is:

The loop stops as temp is 0, and digits is 2.

Variations

1. Kaprekar numbers in a range "Accept two numbers m and n and display all the Kaprekar numbers between m and n."
2. Kaprekar numbers up to a limit "Display all the Kaprekar numbers from 1 to 1000."
3. Using a method that returns boolean "Write a method that accepts an integer and returns true if it is a Kaprekar number, otherwise false."